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Cut-off frequencies are well known in acoustic ducts to be the thresholds of propagation and evanescence. If at one end of a duct the piston oscillates at very near the cut-off frequency, cross-duct resonance occurs and the linearized theory breaks down. This paper studies the nonlinear response, near a cut-off frequency of a guided wave, as an initial-boundary-value problem. The asymptotic state is shown to be governed by a modified cubic Schrodinger equation. Numerical solutions are then obtained for inputs of finite and long duration. In addition to the characteristics of the input envelope, two quantities control the transient phenomenon: frequency detuning and nonlinearity. Under certain circumstances, energy can be trapped near the piston long after a short-lived input has expired, while for a sustained input there is no sign of a steady state. Dissipation is not considered.

A simple formula for ‘wave damping’ is derived, exact within the context of the proposed theory, namely: potential flow correct to second order in the wave amplitude and to leading order in U/c, where U is the drift velocity and c the wave celerity. The analysis is restricted to a two-dimensional problem although the extension to three dimensions seems possible.

During the last decades, as oil production offshore Brazil moved to deeper waters, technical and economical constraints led to a new generation of floating platforms. Nowadays, in the Brazilian offshore scenario, design trends concerning hull form, size and mooring configurations bring novel characteristics of wave-induced dynamics, including non-linear resonant effects. As part of an extensive study on new semi-submersible configurations for Campos basin, recent model tests have shown that their hulls may be subjected to second-order slow motions in heave, pitch and roll. These resonant motions are directly related to the large dimensions and relatively low natural frequencies of the floating systems. The unexpected effects caused great concern, since, in some cases, the low-frequency motions presented amplitudes comparable to those of the first-order response. This paper discusses the evaluation of the 2nd order wave-induced motions of a large-volume semi-submersible platform using WAMIT® second-order module. It is shown that the hydrodynamic forces induced by the 2nd-order potential represent the prevailing effect in the resonant response. Important aspects concerning the numerical model are addressed, such as the parameters involved in the hull and free-surface panelization. Numerical predictions are directly compared with experimental results obtained with a 1:40 model of the platform. A very good agreement is obtained both for heave and angular (pitch or roll) motions, attesting that the numerical code is able to predict the 2nd order forces accurately. Finally, a simplified procedure for dealing with the slow vertical motions is evaluated, aiming to reduce the substantial computational effort required by the 2nd order calculations. Such procedure takes advantage from the fact that the resonant response spectra of the vertical motions are usually narrow-banded (due to the low damping levels) to propose a “white-noise” approach. According to this approach, 2nd order forces need to be calculated only for one frequency difference, corresponding to the natural frequency of the particular motion. Computational time is, therefore, greatly reduced. It is shown that resonant motions calculated through the simplified approach match those predicted through the “full” analysis perfectly, making it an interesting choice for the evaluation of 2nd order effects, especially in the early stages of the design.

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